The low temperature rates for site-to-site transfer of single atoms and molecules adsorbed on surfaces have been determined in recent STM studies@footnote 1@ within the temperature regime where the dominant transfer mechanism changes from mostly activated transmission over to thermally assisted tunneling through the inter-site (transition state) barrier, as the temperature is reduced. A model that has provided useful conceptual and quantitative insights into thermally assisted field emission tunneling spectroscopy@footnote 2@ is used here as the basis for theory of site-to-site atom transfer in this temperature range where proper account of tunneling and quantum reflection, for energies below and above the transition state barrier is required. The predicted transfer rates which are very sensitive to barrier shape as well as height, agree well with those observed in the STM studies of Co and Cu on Cu(111) surfaces in the interesting 4K@<=@T@<=@7K transition range which is relevant in the atom-by-atom fabrication of thermally stable surface nanostructures. @FootnoteText@ @footnote 1@J. Repp, G. Meyer, K.-H. Rieder, and P. Hyldgaard, Phys. Rev. Letters, Vol.91, 206102 (2003); J. A. Stroscio and R. J. Celotta, Science, Vol.306, 242 (2004).@footnote 2@J. W. Gadzuk and E. W. Plummer, Phys.Rev.B Vol.3, 2125 (1971).